Two Ways to Represent Biomarker History
At a landmark time \(s\), a patient’s biomarker history may include several measurements at irregular times, as well as the time elapsed between visits. A prediction model must turn that history into predictors. A landmark model usually does so by selecting or constructing summaries, such as the last observed value, change from baseline, or a recent slope. A joint model instead represents the biomarker through a modeled latent trajectory and links that trajectory to the event process.
These are different choices about predictor information. Neither representation automatically contains every feature that could matter. A summary can discard features that were not selected; a latent-trajectory model can smooth observed values and treat some deviations as measurement error. The relevant question is what information about future risk is retained under the chosen representation and its assumptions.
For illustration, a standard longitudinal submodel has the form \(Y_{ij}=m_i(t_{ij})+\epsilon_{ij}\), where \(Y_{ij}\) is the observed biomarker at time \(t_{ij}\), and \(\epsilon_{ij}\) is measurement error. A common trajectory specification is \(m_i(t)=x_i(t)^\top\beta+z_i(t)^\top b_i\), where \(b_i\) is a subject-specific random effect. A linked event model might specify \(h_i(t\mid b_i)=h_0(t)\exp\{\gamma^\top Z_i+\alpha m_i(t)\}\). Here \(h_0(t)\) is a baseline hazard, \(Z_i\) contains other covariates, and \(\alpha\) describes the association between the modeled biomarker level and the hazard, conditional on the model.
A landmark model at \(s\), by contrast, can model a fixed-horizon event probability or residual survival using \(S_i(s)\) and other predictors, among subjects eligible at \(s\). For either approach, predictions concern a defined target, such as event risk by \(s+u\) given survival to \(s\). A joint-model prediction integrates over uncertainty in the latent effects using the fitted model; it is not generally obtained by substituting one estimated trajectory into the hazard and treating it as known.
What a Selected Summary Retains
A summary is a deliberate compression of history. If a model uses only the last observed value, two histories with the same last value receive the same biomarker predictor even if their earlier levels or trends differ. Adding a baseline value, a change, or a slope can distinguish some of those histories. Each additional feature retains a particular aspect; it does not make the representation equivalent to the full history in every respect.
Worked Example: Same Last Value, Different Trends
Suppose two hypothetical patients have measurements at months 0, 2, and 4. Patient A has values 10, 12, and 14 units; patient B has values 18, 16, and 14 units. Both are event-free at the month-4 landmark, and there are no other differences in the predictors being considered.
A last-value summary gives 14 units for both patients. A model using only that summary therefore has no biomarker-feature distinction between these histories. Their endpoint slopes, however, are different:
A landmark model that includes slope can distinguish the increasing and decreasing histories. A linear latent-trajectory model can also represent different trends, although its estimated slopes depend on the fitted trajectory model and the observed data. The calculations show what each representation can distinguish; they do not establish which patient has higher risk without a fitted risk model and its coefficients.
The same logic applies to nonlinear patterns, variability, and timing. Two patients can have identical last value and endpoint slope but differ in whether the biomarker briefly rose between visits. A summary that omits that feature cannot use it directly. A joint model can use information in the recorded measurements through its specified trajectory likelihood, but only to the extent that the model represents the relevant pattern. For example, a straight-line trajectory cannot describe an unmodeled transient peak as a systematic feature of the latent trajectory.
Why Joint Models Use More Than a Single Estimated Value
In a joint model, the observed history informs the distribution of the latent effects \(b_i\), and therefore the distribution of \(m_i(t)\). For an event-free subject at \(s\), a prediction for survival to \(s+u\) can be written as an average over the latent effects conditional on the available history and survival to \(s\):
Here \(S_i(t\mid b_i,Z_i)\) is survival conditional on the latent effects and covariates, and \(p(b_i\mid H_i(s),T_i>s,Z_i)\) is their conditional distribution for prediction. The survival-to-\(s\) condition matters: in the joint model it can also inform which latent effects are plausible among subjects who remain event-free. Replacing this distribution by a single fitted value discards uncertainty and may alter a nonlinear risk calculation.
Worked Example: Measurement Error and Shrinkage
Consider a hypothetical random-intercept model for a biomarker at a landmark. Write the latent level as \(10+b_i\), with \(b_i\sim N(0,4)\), and suppose one observed measurement is \(Y_i=10+b_i+\epsilon_i\), where \(\epsilon_i\sim N(0,1)\) independently of \(b_i\). For an observed value of 14 units, the measurement-error model does not equate the latent level with 14.
The likelihood gives \(Y_i-10=4\). Normal-normal updating yields posterior variance \(1/(1/4+1)=0.8\) and posterior mean \(4/(1+1/4)=3.2\) for \(b_i\). Thus the posterior latent level has mean \(10+3.2=13.2\) units and standard deviation \(\sqrt{0.8}=0.8944\) units. The estimate is pulled toward the population mean because the model allows measurement error and gives the subject effect a prior distribution.
For a simple hazard association \(\exp\{0.2b_i\}\), the multiplier evaluated at the posterior mean is \(\exp(0.2\times3.2)=\exp(0.64)=1.896\). Averaging the multiplier over this posterior instead gives
The small difference here illustrates that averaging a nonlinear hazard association is not the same operation as evaluating it at a posterior mean. These calculations use the biomarker-only posterior specified in the example; a full event prediction would also condition on survival to the landmark and integrate the survival ratio over the corresponding posterior distribution.
A Justification: When Does a Summary Preserve the History?
A summary preserves all history information relevant to a particular prediction only under a condition about the prediction target. Let \(R_i(H_i(s),Z_i)\) be the conditional probability of the event by horizon \(u\), given survival to \(s\), history, and covariates. If there exists a function \(r\) such that \(R_i(H_i(s),Z_i)=r\{S_i(s),Z_i\}\) for every history under consideration, then the summary contains everything needed to compute that target. Histories with the same summary necessarily have the same conditional risk.
Conversely, if two histories have the same \(S_i(s)\) and \(Z_i\), but their conditional risks differ, no risk model that uses only \(S_i(s)\) and \(Z_i\) can represent both risks correctly. This is a direct consequence of the model receiving identical predictors for those histories: it must assign them the same prediction. The summary might still work well on average, but it has not preserved the history-specific distinction.
This condition is target-specific. A summary might preserve information relevant to one horizon but not another, or to risk under one event model but not another. A sufficiently detailed representation could include the full observed history, but doing so changes the modeling and estimation problem; it does not automatically provide stable or accurate predictions.
Worked Example: Different Residual Patterns, Same Linear Trajectory Information
Consider measurements at times 0, 1, and 2 for two hypothetical patients. Patient A has values \((10,12,10)\); patient B has \((12,8,12)\). Fit an ordinary straight line to each set at these three equally spaced times. The fitted intercept at the middle time is the mean, \(10.667\) units for both. The fitted slope is \((Y(2)-Y(0))/2\), which is 0 units per time unit for both.
The residuals around each fitted constant line are \((-0.667,1.333,-0.667)\) for A and \((1.333,-2.667,1.333)\) for B. Their residual sums of squares are approximately \(2.667\) and \(10.667\) squared units, respectively. Thus the histories differ in their deviations from the fitted line, even though their fitted intercept and slope match.
In a Gaussian linear model with known residual variance and the same design for both patients, the likelihood for the trajectory coefficients depends on the data through the design cross-products and the coefficient-independent residual term. Since both histories have the same fitted intercept and slope, their posterior distribution for those coefficients is the same under the same prior. A joint model using only that linear latent trajectory would not distinguish these histories through different posterior trajectory coefficients. A landmark model using the last observed value would distinguish them: A has 10 units and B has 12 units at time 2.
This is not a claim that the last value is always more informative. It shows that a joint model’s information is structured by its trajectory and error assumptions: departures treated as residual noise need not become event predictors. A landmark model can use a residual-derived feature if it is deliberately included and supported by the data.
Comparing the Information Strategies
| Feature | Landmark model using selected summaries | Joint model using a latent trajectory |
|---|---|---|
| What enters the prediction | Chosen functions of observed history, such as last value, change, or slope | History informs a modeled trajectory and its conditional distribution |
| What can be discarded | Any history feature not represented by the selected summaries | Features inconsistent with, or not represented by, the trajectory and measurement model |
| Measurement uncertainty | Not automatically separated from the observed summary | Represented through the longitudinal model and uncertainty about latent effects |
| Main modeling vulnerability | Omitting a useful summary or fitting an inadequate risk model for the selected predictors | Misspecifying the trajectory, measurement-error, or event-association structure |
The table describes typical implementations, not an absolute boundary. A landmark model can include many features, including fitted trajectory summaries or selected functions of the raw history. A joint model can be extended to represent more complex trajectories. The comparison is therefore not “observed data versus all information”: it is between particular predictor representations and the assumptions used to construct them.
Diagnostics, Limitations, and a Common Mistake
To assess whether the chosen representation is adequate, first ask whether clinically plausible histories are being mapped to identical predictors despite potentially different risks. Compare predictions across subjects with similar summaries but visibly different histories. For a joint model, inspect whether the fitted trajectory captures important patterns in the observed measurements and whether uncertainty is substantial for sparse histories. For a landmark model, check whether the selected features are well-defined for the observed visit patterns and whether adding a feature changes predictions in a reproducible way. These checks do not prove that one representation is correct; they identify where its information choices may matter.
Neither strategy eliminates the need for adequate data. A highly detailed landmark representation can require many parameters and can overfit. A structured joint model can be more stable when its assumptions are reasonable, but can impose misleading smoothing or associations when they are not. Both methods can be affected by limited measurements, censoring, and the population that remains at risk at the landmark. The scientific interpretation should remain predictive or associational unless a separate causal design and assumptions justify causal claims.
- Common mistake: “A joint model uses the whole history, so it must retain every feature and be more informative than a landmark model.”
- Correction: A joint model uses history through a specified probability model. A feature treated as measurement noise or excluded by the trajectory structure may not influence its latent predictor. Conversely, a selected landmark summary can deliberately retain a feature the joint trajectory omits. Compare the representations and assumptions, not just the model labels.
Check Your Understanding
Use the definitions and results above to answer each question. Brief solutions follow the questions.
- Two subjects have the same last observed biomarker value but different observed slopes. Which representation can distinguish them if the landmark model uses only last value? What additional feature would allow it to do so?
- In the normal-normal example, why is the posterior mean latent level below the observed value of 14 units?
- State the condition under which a selected summary preserves all history information relevant to a specified fixed-horizon risk.
- Why might two histories with the same fitted linear trajectory still yield different last-value landmark predictors?
- Does the phrase “joint model uses the history” imply that every observed deviation affects its event prediction? Explain.
Solutions. (1) A last-value-only model cannot distinguish them because its biomarker predictors match. Adding a slope or another feature that captures the trend can distinguish them. (2) Measurement error is allowed, and the prior centers the subject effect at zero; the posterior therefore shrinks the latent estimate toward the population mean of 10. (3) The conditional risk must be a function only of that summary and the other included covariates; histories with identical predictors must have identical target risks. (4) The fitted linear trajectory summarizes the intercept and slope, while the final observed measurement can differ because of residual variation. (5) No. The effect of observations is mediated by the specified trajectory and measurement model; deviations treated as residual noise may not change the latent trajectory or event prediction.